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Lecture 2: Deconfined quantum criticality
T. Senthil(MIT)
General theoretical questions
• Fate of Landau-Ginzburg-Wilson ideas at quantum phase
transitions?
• (More precise) Could Landau order parameters for the phases
distract from the true critical behavior?
Study phase transitions in insulating quantum magnets
- Good theoretical laboratory for physics of phase
transitions/competing orders.
(Senthil, Vishwanath, Balents, Sachdev, Fisher, Science 2004)
Highlights
• Failure of Landau paradigm at (certain) quantum transitions
• Rough description: Emergence of `fractional’ charge and gauge
fields near quantum critical points between two CONVENTIONAL
phases.
- ``Deconfined quantum criticality’’
• Many lessons for competing order physics in correlated electron
systems.
Phase transitions in quantum magnetism
• Spin-1/2 quantum antiferromagnets on a square lattice.
• ``……’’ represent frustrating interactions that can be tuned to drive
phase transitions.
(Eg: Next near neighbour exchange, ring exchange,…..).
Possible quantum phases
• Neel ordered state
Order parameter – staggered magnetic moment vector
Neel ordering can be destroyed at zero temperature
with suitable frustrating interactions
Example of a non-magnetic ground state -
Valence Bond Solid State (VBS)
• Frozen arrangement of singlet
bonds between pairs of local
moments
• Frozen singlet pattern breaks
lattice translation and rotation
symmetry
Discrete Z4 Order Parameter bond
VBS Order Parameter
• Associate a Complex Number
bond
bond
bond
bondbond
Neel-valence bond solid(VBS) transition
• Neel: Broken spin symmetry
• VBS: Broken lattice symmetry.
• Landau – Two independent order
parameters.
- no generic direct second order
transition.
- either first order or phase
coexistence.
This talk: Direct second order
transition but with description not
in terms of natural order
parameter fields.
NeelVBS
Neel +VBS
First order
Naïve Landau expectation
Neel-Valence Bond Solid transition
• Naïve approaches fail
Attack from Neel ≠Usual O(3) transition in D = 3
Attack from VBS ≠ Usual Z4 transition in D = 3
(= XY universality class).
Why do these fail?
Topological defects carry non-trivial quantum numbers!
Attack from VBS (Levin, TS, ‘04 )
Topological defects in Z4 order
parameter
• Domain walls – elementary wall has π/2 shift of clock
angle
Z4 domain walls and vortices
• Walls can be oriented; four such walls can end at point.
• End-points are Z4 vortices.
Z4 vortices in VBS phase
Vortex core has an unpaired
spin-1/2 moment!!
Z4 vortices are spin-1/2
``spinons’’.
Domain wall energy
linear confinement
in VBS phase.
Z4 disordering transition to Neel state
• As for usual (quantum) Z4 transition, expect clock
anisotropy is irrelevant.
(confirm in various limits).
Critical theory: (Quantum) XY but with vortices that
carry physical spin-1/2 (= spinons).
Alternate (dual) view
• Duality for usual XY model (Dasgupta-Halperin)
Phase mode - ``photon’’
Vortices – gauge charges coupled to photon.
Neel-VBS transition: Vortices are spinons
=> Critical spinons minimally coupled to fluctuating U(1) gauge field*.
*non-compact
Critical theory
``Non-compact CP1 model’’
2
4222
)(
|||||)(|
a
zuzrziaxddS
z = two-component spin-1/2 spinon field
aμ = non-compact U(1) gauge field.
Distinct from usual O(3) or Z4 critical
theories*.
Theory not in terms of usual order parameter fields
but involve fractional spin objects and gauge fields.
*Distinction with usual O(3) fixed point due to non-compact gauge field
(Motrunich,Vishwanath, ’03)
Attack from Neel(TS, Vishwanath,Balents, Sachdev, Fisher ’04)
Field theory of quantum antiferromagnets
• Deep in Neel phase (or close
to it) describe by quantum
O(3) non-linear sigma model
field theory.
• Succesful in describing
experiments in cuprate Mott
insulators with Neel ground
states.
fieldparameter order Neelˆ
ˆˆ2
1 222
n
Snng
xddS B
• SB = ``Berry phase’’ sensitive to
microscopic quantized spin at
each site.
• Unimportant in Neel but crucial for
VBS.
Topology of quantum antiferromagnetism
• In d= 2, there are skyrmions in the Neel field.
Topology (contd)
• For the original lattice model, skyrmion number can change (shrink it down to lattice scale and let it disappear).
• In D = 2 + 1, such an ``instanton’’ or ``monopole’’ event is a
hedgehog configuration of the Neel field.
Berry phases and topology
For all smooth configurations of the Neel field, SB vanishes.
However SB finite in the presence of monopoles.
Precise calculation (Haldane ’88): Berry phase factor associated with
each monopole event that oscillates from plaquette to plaquette.
Final result: Only quadrupled monopoles survive!
(Skyrmion number can change but in units of 4).
Field theory of VBS phases
• Sum over different skyrmion number changing events
with Haldane phases (Read-Sachdev).
• Destruction of Neel order through monopole proliferation.
• Haldane phases of monopoles lead to VBS order.
Phase transition – tractable deformations
1. Easy plane anisotropy
2. Embed physical model in a family of models
parametrized by integer N.
N = 2: model of interest.
N = 1 and N = ∞: solvable limits.
Same qualitative picture from all these tractable
deformations
- Quadrupled monopoles irrelevant at transition!!
Critical theory
• Universality class of O(3) model with monopoles
suppressed by hand Early papers:
Dasgupta &Lau ‘89, Kamal & Murthy ‘94
Important recent progress: Motrunich & Vishwanath ’04
Same result as that obtained by attack from VBS!
Quadrupled monopole fugacity = 4-fold clock anisotropy
Renormalization group flows
Clock anisotropy
= quadrupled
Instanton fugacity
Deconfined critical fixed point
Clock anisotropy is ``dangerously irrelevant’’.
Precise meaning of deconfinement
• Z4 symmetry gets enlarged to XY
Domain walls get very thick and very cheap near the transition.
=> Domain wall energy not effective in confining Z4 vortices (= spinons)
.
Formal: Extra global U(1) symmetry
not present in microscopic model :
Two diverging length scales in paramagnet
Lξ ξVBS``Critical” `` spin
liquid’’
VBS
ξ: spin correlation length
ξVBS : Domain wall thickness.
ξVBS ~ ξκ diverges faster than ξ
Spinons confined in either phase
but `confinement scale’ diverges
at transition – hence `deconfined
criticality’.
Analogies to heavy fermion physics
Other examples of deconfined critical points
1. VBS- spin liquid (Senthil, Balents, Sachdev, Vishwanath, Fisher, ’04)
2. Neel –spin liquid (Ghaemi, Senthil, ‘06)
3. Certain VBS-VBS (Fradkin, Huse, Moessner, Oganesyan, Sondhi, ’04; Vishwanath, Balents,Senthil, ‘04)
4. Superfluid- Mott transitions of bosons at fractional filling on various lattices (Senthil et al, ’04, Balents et al, ’05,…….)
5. Spin quadrupole order –VBS on rectangular lattice (Numerics: Harada et al, ’07;Theory: Grover, Senthil, 07)
……..and many more!
Apparently fairly common
A `cool’ new example
Numerical/experimental sightings of Landau-forbidden quantum phase
transitions
Weak first order/second order quantum transitions between two phases with very different broken symmetry surprisingly common….
Numerics
Antiferromagnet – superconductor (Assaad et al 1996)Superfluid – density wave insulator on various lattices (Sandvik et al, 2002, Isakov et al, 2006, Damle et al, 2006))
Neel -VBS on square lattice (Sandvik,
Singh, Sushkov,…………….)
Spin quadrupole order –dimer order on rectangular lattice (Harada et al, 2006)
Experiments:
UPt3-xPdx SC – AF with increasing x. (Graf et al 2001)
Best numerical evidence:
Neel-VBS on square lattice
A sample scaling plot
Emergent XY symmetry for dimer order
Other results: critical quantum phases
(Self Organized Quantum Criticality)
• Stability of critical quantum liquid phases in two dimensions (Hermele, Senthil, Fisher, Lee, Nagaosa, Wen, ‘04)
``Deconfined critical phases” or ``Algebraic spin/charge liquids”
No quasiparticle description of low energy physics!!(Rantner, Wen, ’02)
Huge emergent low energy symmetry/slow power law decay for many distinct orders (Hermele, Senthil, Fisher, ‘06)
Important implications for theories of underdoped cupratesAlgebraic spin liquids: Senthil, Lee, ’05, Ghaemi, Senthil, ’06
Algebraic charge liquids: Kaul, Kim, Sachdev, Senthil,’07
Some lessons-I
• Direct 2nd order quantum transition between two phases with
different competing orders possible (eg: between different broken
symmetries)
Separation between the two competing orders
not as a function of tuning parameter but as a function of
(length or time) scale
Lξ ξVBS``Critical” `` spin
liquid’’
VBS
Loss of magnetic correlations Onset of VBS order
Some lessons-II
• Striking ``non-fermi liquid’’ (morally) physics at critical point between
two competing orders.
Eg: At Neel-VBS, spin spectrum is anamolously broad - roughly due to decay into spinons- as compared to usual critical points.
Most important lesson:
Failure of Landau paradigm – order parameter fluctuations do not capture true critical physics even if natural order parameters exist.
Strong impetus to radical approaches to non fermi liquid physics at
magnetic critical points in rare earth metals (and to optimally doped cuprates).
Outlook
• Theoretically important answer to 0th order question
posed by experiments:
Can Landau paradigms be violated at phases and phase
transitions of strongly interacting electrons?
• But there still is far to go to seriously confront non-Fermi
liquid metals in existing materials……….!
Can we go beyond the 0th order answer?
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